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Chapter 10: Problem 62

If \(G\) is a simple graph with 15 edges and \(\overline{G}\) has 13 edges, how many vertices does \(G\) have?

Short Answer

Expert verified
Graph G has 8 vertices.

Step by step solution

01

Understand the relationship between a graph and its complement

For a simple graph and its complement, the sum of the number of edges in both graphs is equal to the number of edges in a complete graph with the same number of vertices.
02

Express the relationship using an equation

Let the number of vertices in graph G be denoted as n. A complete graph with n vertices has \(\binom{n}{2}\) edges. Thus, the equation \(\binom{n}{2} = 15 + 13 \) represents the total number of edges.
03

Solve for the number of vertices

We have the equation \(\binom{n}{2} = 28\). This simplifies to \(\frac{n(n-1)}{2} = 28\). Solving this equation, we multiply both sides by 2 to get \(n(n - 1) = 56\). Finding the integer solution to this quadratic equation, we get \(8 \) vertices, because \(8 \times 7 = 56\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simple Graph
In graph theory, a **simple graph** refers to an undirected graph without loops or multiple edges between the same pair of vertices. This means each edge connects a unique pair of vertices without repetition. Simple graphs are easy to understand and form the basis for more complex graph structures.
To understand them better:
  • Imagine a set of points (called vertices) connected by lines (called edges).
  • There are no extra lines connecting the same points more than once.
  • No vertex is connected to itself (no loops).
These properties help keep the graph simple and straightforward, ensuring clarity in relationships between vertices.
Complement of a Graph
The **complement of a graph** \(\backslash overline{G}\) involves a transformation of the original graph G. In the complement:
  • Every pair of vertices that are connected by an edge in G are NOT connected in \(\backslash overline{G}\).
  • Every pair of vertices that are NOT connected by an edge in G are connected in \(\backslash overline{G}\).
Essentially, the edges in \(\backslash overline{G}\) represent all the edges missing in G to make it a complete graph. This understanding is crucial for solving problems involving the relationship between a graph and its complement.
Complete Graph
A **complete graph** is a simple graph in which every pair of distinct vertices is connected by a unique edge. Imagine a network where everyone knows everyone else. This type of graph is critical to understand because:
  • It contains the maximum number of edges possible for a given number of vertices.
  • If the graph has n vertices, the number of edges is \(\backslash binom{n}{2}\), which equals \(\frac{n(n-1)}{2}\).
This formula helps in deriving the relationships and counts of edges when considering complements and other graph properties.
Vertices
In graph theory, **vertices** (or nodes) are the fundamental units used to build graphs. They represent points or locations that can be connected by edges. Understanding vertices involves noting:
  • In a graph G, the number of vertices is denoted as **n**.
  • Vertices can be connected by edges to form various structures and shapes.
  • The degree of a vertex is the number of edges incident to it.
Each vertex in a simple graph represents a unique entity that can be connected to other vertices without any repetitions or loops.
Edges
An **edge** in graph theory is a line connecting two vertices. These connections can define many characteristics of graphs. To understand edges, keep these points in mind:
  • In a simple graph, an edge connects two distinct vertices without repetitions.
  • An edge between vertex A and vertex B is unique in a simple graph.
  • In a complete graph, the total number of edges is given by \(\backslash binom{n}{2}\), where n is the number of vertices.
Edges play a crucial role in defining the structure and properties of graphs, making them essential for graph analysis and understanding relationships within data.

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Most popular questions from this chapter

How much storage is needed to represent a simple graph with \(n\) vertices and \(m\) edges using a) adjacency lists? b) an adjacency matrix? c) an incidence matrix?

Show that a directed multigraph having no isolated vertices has an Euler path but not an Euler circuit if and only if the graph is weakly connected and the in-degree and out-degree of each vertex are equal for all but two vertices, one that has in-degree one larger than its out- degree and the other that has out-degree one larger than its in-degree.

A knight is a chess piece that can move either two spaces horizontally and one space vertically or one space horizontally and two spaces vertically. That is, a knight on square \((x, y)\) can move to any of the eight squares \((x \pm 2, y \pm 1),(x \pm 1, y \pm 2),\) if these squares are on the chessboard, as illustrated here. A knight's tour is a sequence of legal moves by a knight starting at some square and visiting each square exactly once. A knight's tour is called reentrant if there is a legal move that takes the knight from the last square of the tour back to where the tour began. We can model knight's tours using the graph that has a vertex for each square on the board, with an edge connecting two vertices if a knight can legally move between the squares represented by these vertices. Show that there is a knight's tour on a \(3 \times 4\) chessboard.

The thickness of a simple graph \(G\) is the smallest number of planar subgraphs of \(G\) that have \(G\) as their union. $$ \text { Show that } K_{3,3} \text { has } 2 \text { as its thickness. } $$

Find the number of nonisomorphic simple graphs with seven vertices in which each vertex has degree two.
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